Heeelp ! Washing machine drain pipe overflowing..What is wrong and how do I fix?

You will have to get that drain line snaked out. Try a plumber or Roto Rooter or something like that. the clog isn't complete, it is letting some water though which is why you can take showers and flush the toilet - not as much water as one time as the washing machine. You can try spinning your washer out, but you will probably have to do it a little at a time until you get most of the water out then you can let it spin the clothes fully and dry them. These things happen in any home, you just have to tackle the problems as they come up. Good luck.